Power plane splitting using a contour method

ABSTRACT

A method for power plane splitting. The method enables the traces on a power plane to be organized so that the conductor area is expanded while still ensuring that components with similar power supply requirements are coupled to the same trace. A potential field of a power plane of a printed circuit board is calculated by assigning one or more potential values to one or more components coupled to the printed circuit board and solving for a plurality of potential field values at a plurality of locations between the one or more components. One or more boundaries between the one or more components are defined by selecting contours of constant potential within the calculated potential field. One or more traces on the power plane are created using the one or more boundaries, wherein the one or more traces connect a corresponding one or more pluralities of components and each plurality of components of the one or more.

TECHNICAL FIELD

This invention relates generally to the field of electronic devices andsystems, and more specifically to circuit board technology.

BACKGROUND

The power plane of a circuit board can accommodate several voltagesources, which supply power to the elements of the circuit board. Thepower plane may be constructed by dividing a large conductor intoseveral distinct, non-shorting regions, where each region connects aspecific voltage source to components of the board. Unfortunately, agiven voltage source may provide power to many locations on the board.This makes power plane design difficult. Manually determining a goodchoice of the non-shorting regions is time-consuming, and can be errorprone. Distinct conductors on a power plane are also referred to aspower nets. If there are any shorted nets on the PCB power layer, thenthe board is useless and must be scrapped.

Automated trace routing is another well-known method for connectingnodes in a circuit board with electrical conductors or traces. Thismethod does not attempt to create traces with a large conductor area. Ona power plane, where electrical current is high, however, a largeconductor area is desirable or necessary, so as to reduce power loss.

SUMMARY

A method for power plane splitting. The method calculates a potentialfield of a power plane of a printed circuit board by assigning potentialvalues to components coupled to the printed circuit board and solvingfor potential field values at locations between the components. Thepotential field is used in defining boundaries between the components byselecting contours of constant potential within the calculated potentialfield. The boundaries are used to create traces on the power plane. Thetraces connect the components so that each component with a samepotential value is coupled to a same power supply.

BRIEF DESCRIPTION OF THE DRAWINGS

The features of the invention believed to be novel are set forth withparticularity in the appended claims. The invention itself however, bothas to organization and method of operation, together with objects andadvantages thereof, may be best understood by reference to the followingdetailed description of the invention, which describes certain exemplaryembodiments of the invention, taken in conjunction with the accompanyingdrawings in which:

FIG. 1 is a flow diagram of a first method for power plane splitting,according to certain embodiments of the present invention.

FIG. 2 is a flow diagram of a second method for power plane splitting,according to certain embodiments of the present invention.

FIG. 3 is a sample grid layout prior to applying the first method forpower plane splitting, according to certain embodiments of the presentinvention.

FIG. 4 is a sample grid layout after applying a contour solver of thefirst method for power plane splitting, according to certain embodimentsof the present invention.

FIG. 5 is a sample grid layout after defining boundaries using the firstmethod for power plane splitting, according to certain embodiments ofthe present invention.

FIG. 6 is a sample grid layout after re-applying the contour solver ofthe first method for power plane splitting, according to certainembodiments of the present invention.

FIG. 7 is a suggested splitting of the power plane after applying thefirst method for power plane splitting, according to certain embodimentsof the present invention.

FIG. 8 is a sample grid layout after applying an auto routingapplication of the second method for power plane splitting, according tocertain embodiments of the present invention.

FIG. 9 is a sample grid layout after applying a contour solver of thesecond method for power plane splitting, according to certainembodiments of the present invention.

FIG. 10 is a sample grid layout after applying several iterations of thecontour solver of the second method for power plane splitting, accordingto certain embodiments of the present invention.

DETAILED DESCRIPTION

While this invention is susceptible of embodiment in many differentforms, there is shown in the drawings and will herein be described indetail specific embodiments, with the understanding that the presentdisclosure is to be considered as an example of the principles of theinvention and not intended to limit the invention to the specificembodiments shown and described. In the description below, likereference numerals are used to describe the same, similar orcorresponding parts in the several views of the drawings.

Referring now to FIG. 1, a flow diagram of a first method 100 for powerplane splitting is shown, according to certain embodiments of thepresent invention. A power plane that is coupled to one or morecomponents may be split into one or more regions, where each region ofthe one or more regions is selected to couple a supply voltage to asubset of the one or more components. This power plane splitting, may beaccomplished by first assigning potential values to the one or morecomponents, hereinafter referred to as one or more nodes (block 110).Note that these potential values are not necessarily the same as thepotential values the nodes will have during operation, but are chosenfor simulation and illustrative purposes. A potential field solver maythen be used to compute a first potential field at a first one or morelocations between the one or more nodes (block 120). The first potentialfield may then be further processed using a contour solver in order tocreate one or more boundaries between the one or more nodes so that theone or more regions are created (block 130). It is noted that the firstmethod may be performed in an automated fashion without userinteraction, or may be performed in a sequence of semi-automated stepswherein a user verifies an outcome of each step in the sequence in orderto create the one or more boundaries.

Thus, the one or more boundaries serve to partition the power plane intothe one or more regions, where a particular region of the one or moreregions is operable to provide all components within that particularregion with a common supply voltage. In certain embodiments of thepresent invention, one or more areas corresponding to the one or moreregions should be expanded by the placement of the one or moreboundaries. It is noted that expanding the one or more areas is operableto increase a current-carrying capacity on a conductor of each region ofthe one or more regions. In certain embodiments of the presentinvention, each boundary of the one or more boundaries is created byplacing the boundary at a plurality of intermediate values between oneor more nodes of the one or more nodes (block 140). The one or moreboundaries so created may then be assigned a potential value of zero(block 150), and the contour solver may then be re-run in order to moreaccurately define a location of the one or more boundaries (block 160).It is noted than in certain embodiments of the present invention, thecontour solver may be run multiple times in order to provide a moreaccurate location for the one or more boundaries. Additionally, a usermay use the location of the one or more boundaries as an aid indetermining where to manually place one or more traces corresponding tothe one or more regions.

An example of the first method is shown in FIG. 3-FIG. 7. Referring nowto FIG. 3, a sample grid layout 300 prior to applying the first methodfor power plane splitting is shown, according to certain embodiments ofthe present invention. The sample grid layout 300 illustrates a nodes(320, 330, 340), wherein the nodes (320, 330, 340) correspond to threedistinct power supply levels. In this example, the nodes (320, 330, 340)are randomly assigned to one of the three distinct power supply levels.A first nodes 320 corresponds to a first power supply level, a secondnodes 330 corresponds to a second power supply level, and a third nodes340 corresponds to a third power supply level. As illustrated in FIG. 3,each of first nodes 320, second nodes 330, and third nodes 340correspond to multiple nodes. The nodes (320, 330, 340) are graphicallyshown in FIG. 3 on contour plot 310. It is noted that although noconnections between nodes (320, 330, 340) are shown in FIG. 3, one ofskill in the art will recognize that one or more of the nodes (320, 330,340) could be initially connected by one or more traces withoutdeparting from the spirit and scope of the present invention. It isfurther noted that the use of three distinct power levels and 17 nodesis exemplary and should not limit the scope of the application.

Referring now to FIG. 4, the sample grid layout 300 after applying acontour solver of the first method 400 for power plane splitting isshown, according to certain embodiments of the present invention. Afterapplying the contour solver to compute the first potential field of thesample grid layout 300, it is evident in FIG. 4 that the distinct nodes(320, 330, 340) have started to become organized according to the powersupply level of the nodes (320, 330, 340). A first region 410corresponds to nodes 320, a second region 420 corresponds to the secondnodes 330, and a third region 430 corresponds to the third nodes 340.

Referring now to FIG. 5, the sample grid layout 300 after defining oneor more boundaries using the first method 500 for power plane splittingis shown, according to certain embodiments of the present invention. Inthis exemplary embodiment, the one or more boundaries are defined alongequi-potential surfaces between the nodes (320, 330, 340). As anumerical example, if the first nodes 320 is assigned a voltage of 50 V,and the second nodes 330 has a voltage of 80 V, a boundary could bedefined along the equi-potential surface with a value 60 V. In a certainembodiment of the present invention, the one or more boundaries arelocated midway between any two nodes of the nodes (320, 330, 340). It isobserved that a comparison of FIG. 5 with the nodes (320, 330, 340)shows that regions 410, 420, and 430 now have boundaries 510.

Referring now to FIG. 6, the sample grid layout 300 after re-applyingthe contour solver of the first method 600 for power plane splitting isshown, according to certain embodiments of the present invention. Theone or more boundaries previously defined in FIG. 5 are used as initialconditions by setting the one or more boundaries to a common value. Incertain embodiments of the present invention, the common value is zerovolts. The contour solver is then re-run using the nodes (320, 330, 340)and the one or more (grounded) boundaries as initial conditions. Thisre-application of the contour solver has shown that one problem of aboundary extending to an edge of the power plane can be adequatelyaddressed by using the one or more boundaries as initial conditions. Itis also noted that the contour solver has enlarged a corresponding area(region 410, 420, 430) of each of the nodes (320, 330, 340) while at thesame time preventing a short from occurring between any of the nodes.The sample grid layout 300 after re-applying the contour solver of thefirst method 600 can, in a certain embodiment of the present invention,be used to manually define a location of the one or more tracescorresponding to the one or more boundaries where the one or more tracesconnect the nodes (320, 330, 340).

Referring now to FIG. 7, a suggested splitting of the power plane afterapplying the first method 700 for power plane splitting is shown,according to certain embodiments of the present invention. FIG. 7illustrates that the one or more boundaries have been used to createthree regions. Region 710 comprises the nodes 320, region 720 comprisesthe nodes 330, and region 730 comprises the nodes 340. It is observedthat in comparison to the regions (410, 420, 430) shown in FIG. 4,regions 710, 720, and 730 clearly define the one or more boundariesneeded to connect the nodes (320, 330, 340) and the one or moreboundaries shown in FIG. 7 do not extend to the edge of the power plane.The one or more boundaries of FIG. 7 illustrate two features of asplitting of a power plane: Nodes coupled to a same power supply arewithin a single boundary, and nodes coupled to different power suppliesare not coupled together.

Referring now to FIG. 2, a flow diagram of a second method 200 for powerplane splitting is shown, according to certain embodiments of thepresent invention. As in the first method, values are assigned to one ormore locations, corresponding to a second one or more nodes, of thepower plane (block 210). An auto-routing program is then used to make aninitial set of connections between the second one or more nodes (block220). This initial set of connections is operable to be used to moreaccurately refine one or more traces, wherein the one or more tracesinterconnects the second one or more nodes so that each node coupled toa certain trace of the one or more traces has a same supply voltagerequirement as any other node coupled to the certain trace of the one ormore traces. An important characteristic of a trace is trace width. Thesecond method may be used to maximize trace widths of conductor areaswithout allowing shorting between adjacent traces.

After running the auto-routing program, virtual potentials are assignedto each of the independent nodes and connecting traces, as before.Again, these potentials are used for simulation, and do not necessarilycorrespond to the potentials required during operation of the PCB. Apotential solver is used to compute a second potential field at a secondone or more locations between the second one or more nodes (block 230).Next, a gradient of the second potential field is computed (block 240).The gradient and the second potential field may then be used to expandan area of a conductor connecting one or more of the second one or morenodes while preventing a short from occurring between one or more of thesecond one or more nodes (block 250). In certain embodiments of thepresent invention, the conductor area may be expanded where both of thefollowing occur:

-   -   1. The second potential field is within a specified percentage        of the conductor potential.    -   2. The value of the gradient is within a specified percentage of        zero.

The process of computing the second potential field, the gradient field,and expanding the area of a conductor (blocks 230, 240, 250) may berepeated until the one or more traces corresponding to a second one ormore boundaries of the power plane are sufficiently wide.

A common value for the potential tolerance is 20% of the difference tothe nearest node potential, so that the area may be expanded when thepotential is within 20% of the conductor potential. A greater value ofthe conductor potential tolerance allows a trace to get wider, morequickly. A common value for the gradient tolerance is 100 ΔV Volts/mil,where ΔV is the difference between the assigned potentials of twoadjacent nodes, so that the area may be expanded when the gradient isless than this threshold. This threshold corresponds to a minimum nodeseparation of 10 thousandths of an inch common industry clearancebetween different conductive regions on a power plane. It is noted thatthese gradient tolerance and potential tolerance are partially driven bythe current state of the art in printed circuit board technology. Theabove stated gradient tolerance and potential tolerance values may beadjusted as PCB technology changes. Note that by making the gradientthreshold larger, different conductors are allowed to come closertogether—a smaller gradient threshold inhibits different conductors fromcoming together. These parameters allow the operator to adjust widthsaccording to their preference.

Referring now to FIG. 8, a sample grid layout after applying an autorouting application of the second method 800 for power plane splittingis shown, according to certain embodiments of the present invention. Thesample grid layout illustrates the second one or more nodes that havebeen coupled into three groupings of nodes. A first nodes 810 areconnected using an auto-routing program. Similarly, a second nodes 820and a third nodes 830 are correspondingly coupled. The auto-routerprogram may be used when there is some initial information onrequirements for connecting the second one or more nodes. The methodillustrated in FIG. 2 may then be used to optimize three conductor areascorresponding to the first nodes 810, second nodes 820, and third nodes830.

Referring now to FIG. 9, the sample grid layout after applying apotential solver of the second method 900 for power plane splitting isshown, where the second nodes 820 has been assigned a potential of 2volts, the first 810 and third 830 nodes have an assigned potential of 1volt, and the edges have an assigned potential of 0 volts, in accordancewith certain embodiments of the present invention. From the sample gridlayout after applying the potential solver of the second method 900,which illustrates a first region 920 surrounding the first nodes 810 andthe third nodes 830 and a second region 910 surrounding the secondnodes, it is noted that a trace connecting the second nodes 820 can beexpanded without causing any of the second nodes 820 to be shorted to anode of the first nodes 810 or a node of the third nodes 830.

The extent to which the traces of the second nodes 820 are expandeddepends upon the potential and gradient thresholds, as specified by theuser. For this example, the potential threshold was 1.6 volts, and thegradient threshold was set at 1 Volt/unit length. The potentialthreshold allows the traces of 820 to expand wherever the voltage isgreater than 1.6 volts, as long as the gradient is less than 1. By notexpanding where the gradient exceeds threshold, we maintain a minimumseparation between nodes of 2 unit lengths on the graph.

Referring now to FIG. 10, a sample grid layout after applying severaliterations of the potential solver of the second method 1000 for powerplane splitting is shown, according to certain embodiments of thepresent invention. The second method, illustrated in the flowchart ofFIG. 2, has been iterated in order to attain three distinct non-shortingregions (1010, 1020, and 1030). Note that in this example, the powerplane has been filled out so that the first nodes 810, the second nodes820, and the third nodes 830 are non-shorting and the three distinctnon-shorting regions (1010, 1020, and 1030) have large respectiveconductive areas. It is noted that a user may further verify that atrace corresponding to a boundary created using the first method or thesecond method has a sufficient width and a sufficient resistance to meetdesign requirements. The verification of the trace may be donegraphically using a plot substantially similar to the sample grid layoutillustrated in FIG. 10.

Certain embodiments of power plane splitting may be applied to printedcircuit boards (PCBs), or to any electronic circuit fabricationtechnology in which multiple power supplies are connected to multiplecomponents. It is further noted that the one or more components may begrouped according to values of power supplies that supply the one ormore components.

Those of ordinary skill in the art will recognize that the presentinvention has been described in terms of exemplary method embodimentsand that these embodiments may be implemented by means ofcomputer-readable media tangibly embodying a program of instructionsexecutable by a computer or programmed processor, as well as hardwareequivalent components such as special purpose hardware and/or dedicatedprocessors. Similarly, general purpose computers, microprocessor basedcomputers, micro-controllers, optical computers, analog computers,dedicated processors and/or dedicated hard wired logic may be used toconstruct alternative equivalent embodiments of the present invention.Such alternatives should be considered equivalents.

The present invention may be implemented using one or more programmedprocessor executing programming instructions that are broadly describedabove in flow chart form and which can be stored in any suitableelectronic storage medium. However, those skilled in the art willappreciate that the processes described above can be implemented in anynumber of variations and in many suitable programming languages withoutdeparting from the present invention. Thus, the process can be describedby a set of instructions implementing the processes described and storedon a computer storage medium such as a magnetic disc, optical disc,magneto-optical disc, semiconductor memory, etc. Many such variationsand modifications are contemplated and considered equivalent.

While the invention has been described in conjunction with specificembodiments, it is evident that many alternatives, modifications,permutations and variations will become apparent to those of ordinaryskill in the art in light of the foregoing description. Accordingly, itis intended that the present invention embrace all such alternatives,modifications and variations as fall within the scope of the appendedclaims.

1. A method for power plane splitting, comprising: calculating apotential field of a power plane of a printed circuit board, byassigning one or more potential values to one or more components coupledto the printed circuit board and solving for a plurality of potentialfield values at a plurality of locations between the one or morecomponents; defining one or more boundaries between the one or morecomponents by selecting contours of constant potential within thecalculated potential field; and creating one or more traces on the powerplane using the one or more boundaries, wherein the one or more tracesconnect a corresponding one or more pluralities of components and eachplurality of components of the one or more pluralities of components iscoupled to a same power supply.
 2. The method of claim 1, furthercomprising assigning a fixed potential value to the one or moreboundaries and re-calculating the potential field of the power plane. 3.The method of claim 1, wherein a user verifies a minimum trace width ofthe one or more traces.
 4. The method of claim 1, wherein a boundary ofthe power plane has a potential value of zero.
 5. The method of claim 1,wherein a potential value of the one or more potential values of the oneor more components corresponds to a voltage of a component of the one ormore components.
 6. The method of claim 1, wherein the one or moreboundaries are selected by using contours having a potential valuesubstantially intermediate to a potential value of a first component anda potential value of a second component.
 7. The method of claim 1,wherein defining the one or more boundaries further comprises computinga gradient of the calculated potential field and using one or moregradient values to determine if two components have a same potentialvalue.
 8. The method of claim 1, wherein the plurality of values ofpotential corresponding to the plurality of locations are operable to bedisplayed graphically, thereby enabling a user to adjust a location ofthe one or more boundaries.
 9. The method of claim 1, wherein the one ormore boundaries are identified using a plurality of contours, eachcontour being defined as a subset of the one or more locations having asame potential value.
 10. The method of claim 1, wherein a gradient ofthe plurality of values of potential corresponding to the plurality oflocations is operable to indicate if a first component has a samepotential value as a second component.
 11. The method of claim 10,wherein the indication that the first component has the same potentialvalue as the second component is operable to prevent shorting the one ormore traces.
 12. A method for automated printed circuit board (PCB)power plane splitting, comprising: calculating a potential field of apower plane of a printed circuit board by assigning one or morepotential values to one or more components coupled to the printedcircuit board and solving for a plurality of potential field values at aplurality of locations between the one or more components; defining oneor more boundaries between the one or more components using contours ofconstant potential within the calculated potential field; assigning afixed potential value to the one or more boundaries; and iterativelycalculating an updated potential field using the one or more potentialvalues of the one or more components and the fixed potential value ofthe one or more boundaries, wherein the updated field is operable to beused to further refine a location of the one or more boundaries.
 13. Themethod of claim 12, wherein the fixed potential value is zero.
 14. Themethod of claim 12, wherein a potential value of the one or morepotential values of the one or more components corresponds to a voltageof a component of the one or more components.
 15. The method of claim12, wherein the plurality of values of potential corresponding to theplurality of locations are displayed graphically for a user, therebyallowing a user to create one or more traces from the one or moreboundaries.
 16. The method of claim 12, wherein defining the one or moreboundaries further comprises computing a gradient of the calculatedpotential field and using one or more gradient values to determine iftwo components have a same potential value.
 17. The method of claim 16,wherein determining if two components have the same potential value maybe used to prevent shorting one or more traces derived from the one ormore boundaries.
 18. A method for printed circuit board (PCB) powerplane splitting, comprising: making one or more initial connectionsbetween a first one or more components coupled to a printed circuitboard employing an auto routing algorithm; calculating a potential fieldof a power plane of the printed circuit board, wherein the potentialfield is calculated by assigning one or more potential values to asecond one or more components coupled to the printed circuit board andsolving for a plurality of potential field values at a plurality oflocations between the second one or more components; computing agradient of the potential field and using the gradient to identify oneor more boundaries, wherein said one or more boundaries are operable todivide the second one or more components into one or more groups whereeach component in a group has a same potential value; expanding a widthof a specific boundary of the one or more boundaries when acorresponding gradient of the specific boundary is substantially closeto a minimal value; and iterating the calculation of the potentialfield, the computation of the gradient, and the expansion of boundarywidths until the one or more boundaries have corresponding widths thatare sufficient to provide a corresponding one or more voltages to theone or more components.
 19. The method of claim 18, wherein the firstone or more components are a subset of the second one or morecomponents.
 20. The method of claim 18, wherein the minimal value iszero.
 21. The method of claim 18, wherein substantially close is definedas having two adjacent nodes within 100 ΔV Volts/mil.
 22. The method ofclaim 18, wherein the one or more boundaries are assigned a fixedpotential value.
 23. The method of claim 18, wherein a user verifies aminimum trace width of the one or more boundaries.
 24. The method ofclaim 18, wherein a potential value of the one or more potential valuesof the one or more components corresponds to a voltage of a component ofthe one or more components.
 25. The method of claim 18, wherein the oneor more boundaries are selected by using contours having a potentialvalue substantially intermediate to a first component potential valueand a second component potential value.
 26. A method for power planesplitting, comprising: dividing a plurality of components coupled to aprinted circuit board (PCB) into one or more groups corresponding to oneor more power supplies, wherein each component of a group is supplied bya same power supply; associating with each group of the one or moregroups a specific value of potential, wherein a plurality of locationson the PCB not corresponding to an element of any group of the one ormore groups comprise a corresponding plurality of values of potential,said corresponding plurality of values of potential being undetermined;calculating the plurality of values of potential corresponding to theplurality of locations using a potential solver; and creating one ormore traces connecting the plurality of components using the pluralityof values of potential, wherein any two components belonging todifferent groups of the one or more groups are connected to differenttraces.
 27. The method of claim 26, further comprising calculating anupdated potential field using the one or more potential values of theone or more groups and a fixed potential of a one or more boundariesbetween the one or more groups, wherein the updated field is operable tobe used to refine a location of the one or more boundaries.
 28. Themethod of claim 26, wherein a potential value of a group corresponds toa voltage of any component of the group.
 29. The method of claim 26,further comprising identifying one or more boundaries of the one or moregroups.
 30. The method of claim 29, wherein the one or more boundariesare identified using contours, each contour being defined as a subset ofthe one or more locations having a same potential value.
 31. The methodof claim 26, wherein a gradient of the plurality of values of potentialcorresponding to the plurality of locations is operable to indicate ifone or more members of one or more groups belong to a same group. 32.The method of claim 31, wherein the indication that one or more membersbelong to the same group may be used to prevent shorting one or moremembers belonging to different groups.
 33. Computer-readable mediatangibly embodying a program of instructions executable by a computer toperform a method for power plane splitting, the method comprising:calculating a potential field of a power plane of a printed circuitboard, by assigning one or more potential values to one or morecomponents coupled to the printed circuit board and solving for aplurality of potential field values at a plurality of locations betweenthe one or more components; defining one or more boundaries between theone or more components by selecting contours of constant potentialwithin the calculated potential field; and creating one or more traceson the power plane using the one or more boundaries, wherein the one ormore traces connect a corresponding one or more pluralities ofcomponents and each plurality of components of the one or morepluralities of components is coupled to a same power supply.
 34. Amethod for power plane, splitting, comprising: means for calculating apotential field of a power plane of a printed circuit board, byassigning one or more potential values to one or more components coupledto the printed circuit board and solving for a plurality of potentialfield values at a plurality of locations between the one or morecomponents; means for defining one or more boundaries between the one ormore components by selecting contours of constant potential within thecalculated potential field; and means for creating one or more traces onthe power plane using the one or more boundaries, wherein the one ormore traces connect a corresponding one or more pluralities ofcomponents and each plurality of components of the one or morepluralities of components is coupled to a same power supply. 35.Computer-readable media tangibly embodying a program of instructionsexecutable by a computer to perform a method for printed circuit board(PCB) power plane splitting, the method comprising: making one or moreinitial connections between a first one or more components coupled to aprinted circuit board employing an auto routing algorithm; calculating apotential field of a power plane of the printed circuit board, whereinthe potential field is calculated by assigning one or more potentialvalues to a second one or more components coupled to the printed circuitboard and solving for a plurality of potential field values at aplurality of locations between the second one or more components;computing a gradient of the potential field and using the gradient toidentify one or more boundaries, wherein said one or more boundaries areoperable to divide the second one or more components into one or moregroups where each component in a group has a same potential value;expanding a width of a specific boundary of the one or more boundarieswhen a corresponding gradient of the specific boundary is substantiallyclose to a minimal value; and iterating the calculation of the potentialfield, the computation of the gradient, and the expansion of boundarywidths until the one or more boundaries have corresponding widths thatare sufficient to provide a corresponding one or more voltages to theone or more components.